The Moyal-Lie Theory of Phase Space Quantum Mechanics

T. Hakio˘glu Physics Department, Bilkent University, 06533 Ankara, Turkey and Alex J. Dragt Physics Department, University of Maryland, College Park MD 20742-4111, USA

Abstract

A Lie algebraic approach to the unitary transformations in Weyl quantization is discussed. This approach, being formally equivalent to the ?-quantization is an extension of the classical Poisson-Lie formalism which can be used as an efficient tool in the quantum phase space transformation theory. The purpose of this paper is to show that the Weyl correspondence in the quantum phase space can be presented in a quantum Lie algebraic perspective which in some sense derives analogies from the Lie algebraic approach to classical transformations and Hamiltonian vector fields. In the classical case transformations of functions f(z) of the phase space z =(p, q) are generated by the generating functions µ(z). If the generating functions are elements of a set, then this set usually contains a subsetG closed under the action the Poisson Bracket (PB), so defining an algebra. To be more precise one talks about different representations of the generators and of the phase space algebras. The second well known representation is the so called adjoint representation of the Poisson algebra of the ’s in terms of the classical Gµ Lie generators L µ over the Lie algebra. These are the Hamiltonian vector fields with the G Hamiltonians µ. By Liouville’s theorem, they characterize an incompressible and covariant phase space flow.G The quantum Lie approach can be based on a parallelism with the classical one above. In this case the unitary transformations are represented by a set of quantum generating functions µ(z) which are the quantum partners of the µ’s. The non commutativity is encoded A G ˆ in a new multiplication rule, i.e. ?-product in z. The quantum Lie generators V µ over A the Lie bracket are the adjoint representations of the generating functions µ(z)overthe Moyal bracket and they correspond to quantum counterparts of the classicalA Hamiltonian vector fields L µ . They are represented by an infinite number of phase space derivatives. G The only exception is the generators of linear canonical transformations defining the sp2(R) sub algebra. In this case the generators are given by the classical Hamiltonian vector fields generated by quadratic Hamiltonians and the classical and quantum generators are identical [see (46) below]. The classical Lie approach is sometimes referred to as the Poisson-Lie theory (PLT). This terminology is quite appropriate for generalizations as the algebraic representations that are relevant are written by using the Poisson and Lie product rules. For a similar reason we suggest that the quantum approach be referred to as the Moyal-Lie theory (MLT). In section 1 we start with a brief textbook discussion of the PLT. The Lie algebraic treat- ment of the quantum transformation theory starts in section 2 with a discussion of Weyl correspondence (section 2.1) and the ?-product. The main results of this paper, the MLT and the ?-covariance are discussed at length in section 2.2. The associativity property is discussed comparatively with respect to Poisson-Lie and the Moyal-Lie algebras in section 3. In the light of the classical covariance versus ?-covariance of the phase space trajectories under canonical transformations the time evolution deserves a specific attention. The time evolution in the MLT and its extended ?-covariance property is examined in section 4. Finally, we discuss the equivalence of the MLT to the standard ?-quantization through the ?-exponentiation in section 5.

A. The Poisson-Lie Theory and the classical phase space

The classical Lie generators L µ are given by G T L µ =(∂z µ) J∂z , (1) G G

where µ = µ(z)’s are in the set of phase space functions (generating functions) where µ =1,...,NG Gdescribes the generator index, N being the total number of generators, z denotes the 2n dimensional phase space row vector z =(z1,...,z2n)=(q1,...,qn; p1,...,pn), ∂ =(∂ ,...,∂ ), T is the transpose of a row vector, and J is the 2n 2n symplectic z z1 z2n × matrix with elements (J)ij ,i,j=(1,...,2n) 1 if j=i+n (J)ij = 1 if i=j+n . (2)  −  0 all other cases The classical algebra of the phase space generating functions is deﬁned over the Poisson bracket (PB)

, (P ) =(∂ )T J (∂ )(3) {Gµ Gν} z Gµ z Gν

and that of the generators L µ over the Lie Bracket (LB) G

(P ) [L µ ,L ν ] L µ L ν L ν L µ = L µ, ν (4) G G ≡ G G − G G {G G } where the LB is deﬁned by the ﬁrst equality. A classical canonical transformation (CCT) is considered to be a symplectic phase space map ~ : z Z~(z)withZ =(Q1,...,Qn; P1,...,Pn) describing the new set of canonical phaseM space pairs.7→ The symplectic condition is the invariance of the PB given by

z ,z (P ) = Z ,Z (P ) = J (5) { j k} { j k} jk

which implies that ~ is invertible. If the symplectic map is continuously connected to the identity within a domainM ~ = ~0, i.e. = I, Lie’s first theorem1 states that the first order M~=~0 infinitesimal change in zj is

(1) δj(z, δ )=δµ L µ zj . (6) G 2 where a summation over the generating function index µ is assumed. Finite canonical transformations can be obtained by inﬁnitely iterating Eq. (6) to an exponential operator as

Zj =exp µ L µ zj (7) { G } (P ) and following [L µ ,f]= µ ,f and considering f = zj, G k {G k } 1 Z = z + ,z (P ) + , ,z (P ) (P ) + ... j j µ {Gµ j} 2! µ1 µ2 {Gµ1 {Gµ2 j} } 1 + ... , ..., ,z (P ),... (P ) + ... (8) k! µ1 µk {Gµ1 { {Gµk j} } where summations over the repeated indices are assumed. Let us consider the case N =1. This allows us to simplify the notation and drop the index µ. In fact, the generalization for finite N is nontrivial because of the possibility of additional structures in the Poisson algebra, for instance, PB of the generators µ,whereµ =1,...,N, may be closed under a specific algebra with N generators. For whatG we present here such algebraic generalizations will not be needed. It can be shown that, if f(z) has a convergent Taylor expansion in powers of z in some domain z , the transformation of f is ∈D f(Z)=exp L f(z)exp L (9) { G} {− G} where it is to be noted that Eq. (9) is an operator relation, i.e. it can be multiplied on the left and/or right by any arbitrary function g(z) without changing its validity. The transformed variable Z and f(Z) must also be well defined in the same domain . From Eq. (9) it follows the remarkable manifest covariance property2,3 D

[exp L f](z) f(z)=f([exp L z]) = f(Z(z, )) , (10) { G} ≡ { G } and the generalized Leibniz rule

exp L [f(z) g(z)] = [exp L f][exp L g]=f(Z) g(Z) (11) { G} { G} { G} which can be summarized in

exp L [f(z) ...]=[exp L f(z)] [exp L ...]=f(Z)[exp L ...] . (12) { G} { G} { G} { G} In Eq’s (10-11) the square brackets indicate that the operator on the left acts on the object within the brackets only. In (12) the dots indicate the possibility of further functions to be acted upon. Although Eq’s (10-12) are diﬀerent ways of writing Eq. (9) we will keep them explicitly for that they facilitate a comparison with their quantum analogs. The MLT is a generalization of the Poisson-Lie theory to the non commutative phase space. It will be shown that, the Moyal-Lie generators can be uniquely derived in the quantum case with similar covariance properties when the role of the ordinary functional product is taken by the ?-product. As it turns out, the manifest covariance property in Eq. (10) is modiﬁed4 because of the non commutativity of the ?-product. As opposed to the standard covariance in (10) under ordinary multiplication the new covariance rule is manifest under the ?-multiplication of the functions of the phase space. The classical covariance is then recovered in theh ¯ 0 limit of the ?-covariance. → 3 B. Quantum Lie generators

The quantum mechanical results presented here are corollaries of a recent work of one the present authors4 on the extended covariance in non commutative phase space under canonical transformations5. Our formulation here is based on Weyl correspondence6–8 as a consistent and analytic principle between the classical and the quantum phase spaces.

1. The Weyl correspondence

Letz ˆ =(ˆz1,...,zˆ2n)=(ˆq1,...,qˆn;ˆp1,...,pˆn) describe the phase space operators satisfying, as usual, [ˆzi, zˆj]=ihJ¯ ij and zero for all other Lie Brackets (commutators). We adopt the same deﬁnitions for z and ∂z as in the classical case. The Weyl map is a correspondence rule between the operators described by ˆ = (ˆz) and the functions f(z) denoted by ˆ f and expressed as F F F⇔

ˆ = dµ(z) f(z) ∆(ˆ z) ,f(z)=Tr ˆ ∆(ˆ z) (13) F Z {F } n where dµ(z)= k=1 [dqk dpk/(2πh¯)]. We adopt the same deﬁnitions for z and ∂z as in the ˆ classical case. HereQ ∆(z) describes a mixed basis for operators and functions in the phase space given by

iωT Jz/¯h iωT J z/ˆ ¯h ∆(ˆ z)= dµ(ω) e− e (14) Z 2n where ω =(ω1,...,ω2n) is in the same domain as z, i.e. R , dµ(ω)= n ˆ k=1 [dωk dωk+n/(2πh¯)]. For details on the properties of the continuous ∆(z) basis we refer theQ reader to Ref. [4]. At a fundamental level the Weyl correspondence can be stated as

ei(ωJzˆ)/¯h ei(ωJz)/¯h (15) ⇔ for all real ω. From Eq. (15) the association for the identity, i.e. I 1 is unique for ω =0. All other maps between the operators and the functions in the Hilbert↔ space are derived from Eq. (15) by suitable diﬀerential operations. Eq. (15) is an explicit rule which leaves the algebraic properties of operators in the phase space invariant. Let ˆ f and ˆ g be associated by the Weyl correspondence. Then F↔ G↔

ih¯ (∂← J ∂→ ˆ ˆ f?g= fe2 z z ) g (16) F G↔ where ? is a non commutative, i.e. f?g g?f = 0, and associative, i.e. f?(g?h)=(f?g)?h operation corresponding to the phase− space symbol6 of the operator product. One way to deﬁne the ? product is by its action explicitly on the functions of the phase space by ih¯ ih¯ f? = f(z + J ∂→ ) ,?f= f(z J ∂← ) (17) 2 z − 2 z where the arrows indicate the action of the derivatives. We explicitly write the correspondence for the LB as

4 [ ˆ, ˆ] f?g g?f= f,g (M) (18) F G ⇔ − { } where the superscript M denotes the Moyal Bracket (MB) deﬁned by the well-known ex- pression

f,g (M) = ih¯ f,g (P ) + (¯hk,∂(k)) . (19) { } { } O z 3 k ≤

Thefirsttermisih¯ times the PB as defined in Eq. (3). (¯ h3,∂(3)) represents higher order O z terms which appear in infinitely odd powers of bothh ¯ and ∂z.

2. The Unitary transformations in phase space and quantum Lie generators

All invertible transformations (including non-unitary ones) can be formalized in the quantum phase space4. In this paper we will only consider the unitary ones. Let ˆ be an operator iγ ˆ F and Uˆ = e A a one parameter unitary transformation with γ R. In the following we proveA that the unitary transformation of ˆ by Uˆ in the operator∈ space has a unique representation in the phase space given by4 F A

iγ Vˆ ˆ0 = Uˆ † ˆ Uˆ f 0(z)=e A f(z) (20) F A F A ⇔ where, ˆ f, ˆ0 f 0 as usual, and Vˆ is a nonlinear and Hermitian generator associatedF⇔ with a realF phase⇔ space generating functionA (z)byVˆ = ? ? . Since ˆ is considered to be Hermitian, it can be representedA inA a symmetricallyA − A ordered series inA z ˆ as,

ˆ n m r = (ˆz)= an,m,r pˆ qˆ pˆ (21) A A n,m,r X where an,m,r are some real coeﬃcients and we indicated thep, ˆ qˆ dependence explicitly for clarity. In order to prove Eq. (20) we start with

( iγ)k ˆ0 = ˆ iγ [ ˆ, ˆ]+...+ − [ ˆ,...,[ ˆ, ˆ],...] (22) F F− A F k! A A F k and calculate the commutator in the (γ) term. We| use a particular{z } phase space representation ofz ˆ known as the Bopp shiftsO and given by10,11,4 ih¯ ih¯ zˆ∆(ˆ z)=[z J ∂→ ] ∆(ˆ z) , ∆(ˆ z)ˆz =[z + ∂→ ] ∆(ˆ z) (23) − 2 z 2 z

zˆL zˆR where, explicitly | {z } | {z }

i¯h → i¯h → pˆL p 2 ∂q pˆR p + 2 ∂q zˆL = = − , zˆR = = . (24) qˆL! q + i¯h ∂→ ! qˆR! q i¯h ∂→ ! 2 p − 2 p It is readily veriﬁed that

5 [ˆzjL , zˆkR ]=0, [ˆzjL , zˆkL ]=ihJ¯ jk =[ˆzkR , zˆjR ] . (25)

By repeatedly applying Eq’s (23) to the commutator of Eq. (21) with ∆,ˆ one explicitly obtains

ˆ ˆ ˆ ˆ ˆ r m n n m r [ , ∆] = V ∆ , V = an,m,r pˆLqˆL pˆL pˆRqˆR pˆR . (26) A A A n,m,r − X n o Remembering that ˆ in Eq. (21) is symmetrically ordered, Eq. (26) implies for an arbitrary phase space functionAf(z)

ˆ (M) V f =[ (ˆzL) (ˆzR)] f = ?f f? = ,f (27) A A −A A − A {A } where Eq. (17) is used in the last part. Note that Vˆ is an operator, composed of powers of z, ∂ in the same way as ˆ depends on the Bopp operatorsA z ˆ , reproducing the action of z A L,R the commutator [ ˆ, ∆]ˆ in the phase space. By recursive application of (26) the k’th order commutator in Eq.A (22) is represented by

[ ˆ,...,[ ˆ, ∆]ˆ ,...] =(Vˆ )k ∆ˆ , for all 0 k. (28) A A A ≤ k Exponentiating Eq.| (28) {z }

iγ Vˆ ˆ† ∆ˆ ˆ = e− A ∆ˆ . (29) UA UA

We now use this result in the second equation in (13) for ˆ0 f 0 F ⇔ iγ Vˆ f 0(z)=Tr ˆ0 ∆ˆ = Tr Uˆ † ˆ Uˆ ∆ˆ = Tr ˆ Uˆ ∆ˆ Uˆ † = e A f(z) . (30) {F } { A F A } {F A A} producing the correspondence in (20). Note that the unitary transformation of ˆ involves left and right multiplications by the transformation operator Uˆ and its conjugate.F Because of the underlying associativity these operations commute. TheA same is observed also with the exponential phase space operator in Eq. (30). Using Eq. (27) we write

iγ Vˆ iγ [ (ˆzL) (ˆzR)] iγ (ˆzL) iγ (ˆzR) iγ (ˆzR) iγ (ˆzL) e A = e A −A = e A e− A = e− A e A . (31)

Eq. (30) is the simplest form of a canonical transformation acting on the functions of the phase space. Using Eq’s (17) and (27) it can be put in the form

iγ Vˆ iγ [ (ˆzL) (ˆzR)] e A f = e A −A f (iγ)2 = f + iγ ( ?f f? )+ ( ? ?f+ f? ? 2 ?f? )+... A − A 2! A A A A− A A (iγ)2 (iγ)k = f + iγ ,f (M) + , ,f + ...+ ,..., ,f ... + ... {A } 2! {A {A }} k! {A {A } } k iγ ( ?) iγ (? ) = e A fe− A | {z } (32) which can also be deduced from (31) using (25). There is a unique correspondence between Vˆ and the unitary transformation Uˆ that it represents as illustrated by A A

6 f(z) Weyl ˆ iγ Vˆ ⇐⇒ ˆ F e A U (33) m A m iγVˆ Weyl ˆ f 0 = e A f 0 . ⇐⇒ F In particular, if we consider two such transformations acting on the phase space, denoting ˆ the correspondence by 1 V 1 and similarly for 2, we ﬁnd A ←→ A A

ˆ ˆ ˆ (M) [V 1 , V 2 ]=V 1, 2 . (34) A A {A A } ˆ Furthermore, if the operators Ak are the generators of an abstract Lie algebra the corre- ˆ sponding phase space operators V k generate the adjoint representation in the phase space of the same Lie algebra. A ˆ It is clear that V k ’s are quantum analogs of the classical Lie generators L k of classical canonical transformations.A To demonstrate this we examine the quantumG analog of the classical covariance property in Eq. (9-10). Since the analog of is the exponentiated generator in Eq. (30) we start with that equation. Using the correspondenceM in Eq. (16) we ﬁnd

f 0 ?g0 = Tr Fˆ0 Gˆ0 ∆ˆ = Tr Uˆ † Fˆ Gˆ Uˆ ∆ˆ =(f?g)0 . (35) { } { } This equation is the quantum analog of (11) written more explicitly as

iγ Vˆ iγ Vˆ iγ Vˆ [e A f] ? [e A g]=[e A f?g] . (36) where the square brackets have the same role as in (10)-(12). Considering (36) for arbitrary g the analog of Eq. (12) is found as

iγ Vˆ iγ Vˆ iγ Vˆ [e A f] ?e A = e A f? (37)

Eq. (37) is converted into

iγ Vˆ iγ Vˆ f 0? = e A f?e− A (38) which is the quantum analog of Eq. (9). We now examine the covariance properties under iγ Vˆ the action of e− A in analogy with Eq. (10). Suppose that f(z) is expanded in terms of binomials pm qn as

m n f(z)= fm,n p q (39) 0 (m,n) ≤X We use in (39) the fact that pn+m = pn pm =[pn ?pm] for all m, n and similarly for powers of q. Hence f in Eq. (39) is equivalently expressed as

f(p, q)= fm,n [p?...?p] [q?...?q] . (40) 0 (m,n) ≤X m n Note that, in (40), the ordinary product is| neither{z } commutative| {z } nor associative with the ?-product. It can now be checked by using Eq’s (30) and (38) that classical covariance described by Eq. (10) is violated

7 iγ Vˆ f 0 =[e A f](z) = f(Z) (41) 6 where Z = Z(z) are the new canonical coordinates given by

iγ Vˆ Z =[ A z] . (42)

We now trace back to the origin of the violation of the classical covariance in Eq. (41). To illustrate the point we adopt two distinct and simple cases. As the first example let us consider f = pq. Our goal is to look for what causes the effect f 0 = PQ.Byusingthe 6 properties of the ?-product we first write f = p?q,pq+ih/¯ 2. We now apply the transformation in Eq. (30). By Eq’s (35-38) we find f 0 = P? Q + ih/¯ 2. The only way in which pq PQ q,p 7→ under all such transformations as in (42) is when ?q,p = ?Q,P . If this was true the covariance would have been manifest, since we would have had P? Q = P? Q = PQ ih/¯ 2. Then q,p Q,P − inserting this in f 0, we would have found f 0 = PQ. As the second example we consider 2 2 f(p, q)=p =[p?p]. Repeating the same calculation here we find that f 0 = P?q,p P = P . The equality, which indicates covariance, would have been obtained for all transformations6 Q = Q(z)andP = P (z) if and only if ?q,p = ?Q,P . These two typical examples demonstrate that the very intrinsic property of the ? operation, non invariance under a general canonical transformation, i.e. ?q,p = ?Q,P , is responsible for the loss of classical covariance in the quantum case. One exceptional6 case in which ?q,p = ?Q,P can be identified corresponds to the group of linear canonical transformations, i.e. Sp2(R)asgivenby

P p ab = g ,g= Sp (R) . (43) cd 2 Q! q! ∈ By conjugation,

∂P T 1 ∂p T 1 d b =(g )− , (g )− = − . (44) ∂ ∂ ca Q! q! −

By direct inspection of Eq. (44), and (16), the invariance of the ? product, i.e. ?q,p = ?Q,P , is guaranteed by the constant coefficients. We now demonstrate, for the linear canonical transformations, the manifestation of the classical covariance in Eq. (10) and for a more m n general case f(p, q)=p q . By using the associativity property of ?q,p we note that m n m n k p q =[p? ][q? ]where[x?]=[x?x?...?x]with? = ?q,p. Since mixtures of the ? and the k ordinary product is not associative, one must perform them in the order which is denoted | {z } by the square brackets above. Namely, we first perform the calculations within each square bracket and then multiply them by using the ordinary multiplication. Denoting a generator ˆ ˆ ˆ m iγVΩ 1 iγVΩ n of the linear canonical transformation Ω by VΩ, we find that f 0 = P? ? e ?− e− Q? iγVˆΩ 1 iγ{VˆΩ } { }1 wherewehave? = ?q,p.Since?q,p = ?Q,P we have ? e ?− e− =1.Here?− is defined as the inverse of the star product which is{ well defined} { and corresponds} to the complex conjugate of ?. We therefore have f =[P m ][Qn ]=P m Qn for all nonnegative 0 ?P,Q ?P,Q m, n. Hence, under linear canonical transformations we have the desired covariance

ˆ [eiγ VΩ f](z)=f(Z) (45)

8 and all other classical properties in Eq’s (10-9) can be easily derived for the linear canonical transformations here. Note that, the three quantum generators of the linear canonical algebra sp2(R)aregivenby ih¯ 1 2 2 ˆ 1 = (ˆp qˆ ) , V 1 = (p∂q + q∂p) A − 4 − 7→ A 2 ih¯ 1 2 2 ˆ 2 = (ˆp +ˆq ) , V 2 = − (p∂q q∂p) (46) A 4 7→ A 2 − 1 ˆ ih¯ 3 = (ˆp qˆ +ˆq pˆ) , V 3 = (p∂p q∂q) . A 4 7→ A 2 − The corresponding classical generators L are generated by the same quadratic polynomials Gk k in (46). It is observed that the quantum generators are identical to the classical generators (upA to an ih¯ factor), i.e. Vˆ = ihL¯ . In ref. [4] three types of quantum phase space Ak Gk generators, i.e. the generators of sp2(R), the gauge and the contact transformations, have been studied. For all three cases exact solutions are possible and the quantum generating functions are identical to the classical ones. These three generators therefore generate the same canonical algebra and the group of transformations in the classical and quantum phase spaces. No other class of transformations have been found yet having this universal property between the classical and quantum cases. In the light of these possibilities, it was suggested that the canonical algebra is spanned solely by these three classes of generators9 which is an important issue yet to be settled. The time evolution as a canonical transformation, can also be represented in terms of the MLT. Like in the classical case, the Hamiltonian function is the generator of time translations in the phase space. This will be examined in section 4. We now examine the question of associativity in the classical and the quantum phase space algebras.

C. Associativity and classical phase space

The Poisson bracket as an algebraic extension of the abstract Lie bracket is not associative1. One way to understand this is to construct a Poisson product which can be deﬁned as × 1 = ∂← J ∂→ (47) × 2 z z so that the Poisson bracket is represented as

g,f (P ) = g f f g. (48) { } × − × The -product satisﬁes the left and right distributive and the scalar laws. Hence it is a linear× algebra. Nevertheless it is non associative, i.e. g (f h) =(g f) h which can be checked quite easily. × × 6 × × Why is associativity so important in classical phase space? In pre quantum mechanical times the answer to this question would be in order to understand the classical phase space as a group manifold on which a theory of Lie transformations can be established.Anumber of attempts in this direction have been made in the past but the most serious considerations started after the advent of quantum mechanics and were inspired by it1,12.

9 From the classical point of view, the quantum mechanical ?-product can be realized as an associative non-commutative andh ¯ parameterized deformation of the -product in Eq. (47). We may write the ? product in terms of the Poisson product as, ×

? =exp ih¯ . (49) { ×} The associativity of the ?-product is clear by its analytic correspondence with the Lie bracket. It can be illustrative to demonstrate how associativity is manifested in the phase space directly. Using Eq. (25) we note that, the left and right multiplication in ?-product deﬁned by Eq. (17) commute as

[f?,?g]=f(ˆz ) g(ˆz ) g(ˆz ) f(ˆz ) = 0 (50) R L − L R for any two functions f and g. Now consider three such functions f,g,h as

[f?h] ?g= g(ˆzL) f(ˆzR) h = f(ˆzR) g(ˆzL) h (51)

where we used (50) in the last step in (51). The last equality implies that the diﬀerential operator f(ˆzR) acts on all functions on its right

f(ˆzR) g(ˆzL) h = f(ˆzR)[g(ˆzL) h]=f(ˆzR)[h?g]=f?[h?g] (52)

Associativity of the ?-product is then proved by a comparison of Eq’s (51) and (52). However in the case of Poisson product the classical version of (50) is given by [f , g] = 0 for arbitrary f,g causing Eq’s (51) and (52) to be invalid. × × 6 The associativity of the ?-product, combined with its manifest covariance under unitary transformations described by Eq. (35) demonstrates that the ?-product is the algebraic partner in the quantum phase space of the ordinary functional product in the classical phase space8. The coexistence of the ? and ordinary products such as in (40) is very commonly encountered when the unitary transformation of a function f(z) is to be calculated. It was proved in Eq’s (45) that, with the exception of sp2(R) an arbitrary transformation does not preserve the classical covariance property. Here, we have an example of that from time evolution.

D. ?-covariant time evolution in the phase space

In the standard formulations of the quantum mechanics the transformations can be represented by their actions on the states or on the operators or some consistent mixture of both. For the speciﬁc case of the time evolution these are known as the Schr¨odinger, Heisenberg and interaction pictures respectively. We argue in the following in favor of adding to this gallery the phase space picture. One distinction of the phase space picture with respect to the other pictures is that it is essentially an operator picture. In this operator picture the quantum states in the Hilbert space that the phase space operators act on are represented by the density operator. The second distinction is that every admissible phase space operator (see section 2.1) is uniquely mapped onto an admissible phase space function and the operator product is uniquely mapped onto the ?-product. In this scheme, the density operator is uniquely mapped to a Wigner function. In the positivist point of view, the second

10 distinction greatly facilitates the use of the conceptual tools to see the quantum dynamics as a classical dynamics on a deformed phase space4,8. On the other hand, the opposite view is also possible. A negativist view can be based upon concentrating on the difficulties in the computations imported by the new rule of functional multiplication, i.e. ?-product. As it turns out, in the case when a transitive action of a transformation is involved, the classical and quantum cases significantly differ. A typical example is the time evolution. The time evolution in the phase space is a clear picture allowing a comparative analysis of the classical and quantum dynamics given by the same Hamiltonian (z). In the quantum Ht case we assume that t(z) can be mapped to a quantum mechanical Hamiltonian operator ˆ H ˆ t through the Weyl correspondence, i.e. t(z) t. An arbitrary function f of the H H(c`) ⇔ H(q) dynamical variables z will be denoted by ft (z)andft (z) as the classical and quantum solutions corresponding to f. To facilitate the comparison further, we also assume that at (c`) (q) a certain initial time t0 =0,f0 (z)=f0 (z)=f0(z). The classical time evolution can be calculated as

t (c`) dt0 L (c`) ft (z)= e 0 Ht0 f0 (z) . (53) T R where is the time ordering operator deﬁned in the standard way by T

t t t t0 dt0 L 1 2 e 0 Ht0 =1+ dt0 L + dt0 L dt00 L ... (54) Ht0 2! Ht0 Ht00 T R Z0 Z0 Z0 ˆ If Eq. (20) is considered for the quantum time evolution generated by U t , then by Eq’s (26)- (29), the ﬁnite time evolution is represented in the phase space by H

t (q) i ˆ h¯ dt0 V ˆ ft (z)= e− 0 Ht0 f0(z) , V = t ? ? t . (55) Ht0 0 0 T R H − H For Hamiltonians quadratic in z Eq’s (53) and (55) yield identical results. For such ˆ quadratic Hamiltonians L = ih¯ V and hence the time evolution is covariant, i.e. ft(z)= i t H H dt0 Vˆ f ( e− h¯ 0 Ht z). The diﬀerence arises when the third and higher order terms in z are 0 T 0 present inR the Hamiltonian and/or in f0. For example consider the typical case of the quartic oscillator = + λ q4 where H =(p2 + q2)/2. Using (55) we have H H0 4 0 ˆ 3 h¯ 2 V = ih¯ (p∂q q∂p)+i hλ¯ q ∂p q∂p (56) H − − − 2 h i There is no known method of calculating the exact analytic forms of the time dependence of the phase space trajectories generated by Eq. (56). The series expansion generates highly nonlinear polynomials in increasing power of z at each order. A ﬁnite series of arbitrarily high orders can, in principle, be performed numerically for the binomials pm qn. Exact results can be achieved for the type of Hamiltonians = p2/2+V (q) if a periodic kick is introduced in the interaction V (q). The model we considerH is the periodically kicked Hamiltonian

p2 (z)= + λ (q) δ (t) (57) H 2 V T where δT (t) is the periodic delta-kick function with the period T . The quantum case for (57) is exactly solvable for arbitrary potentials V (q) and the solution is identical to the classical

11 case. We start with the classical one. The transformation induced by Eq. (57) in one full step is given in terms of a time ordered product of two unitary Lie generators as

( ) TLp2/2 λL ( ) fn−+1(z)=e e V fn− (z)

TLp2/2 λL ( ) = f(e e V z)=f(zn−+1) (58)

where the classical covariance condition is explicitly stated in the second part of the equation. The minus sign in the superscript indicates that the value of fn is calculated inﬁnitesimally before the n’th kick. The time dependence for zn is given by the standard result

( ) ( ) ( ) p˜ − =˜p − + κ 0(q ) ,q= q p˜ − , (59) n+1 n V n+1 n+1 n − n

withp ˜n = Tpn and κ = Tλ. The quantum analog of Eq. (58) in the operator space is the time ordered quantum evolution operator

2 i pˆ i λ (ˆq) Uˆ = e− h¯ 2 e− h¯ V , (60) H where (ˆq) is the operator corresponding to the potential (q). The unitary Lie generators are determinedV by Eq. (55). Time ordering brings complicationsV and therefore we explicitly derive Vˆ from Eq. (20) and Eq. (29). Consider Eq. (30) and = therein, where is given byH (57). Note that, because of the unitarity of the time evolutionA H and the cyclicityH of the trace, the time ordering is eﬀectively reversed for ∆ˆ by one round of cyclic permutation of the operators in (30). After permuting the Uˆ one round, we have H 2 2 i λ (ˆq) i pˆ i pˆ i λ (ˆq) ˆ† ∆ˆ ˆ = e h¯ V e h¯ 2 ∆ˆ e− h¯ 2 e− h¯ V H H U U i ˆ i ˆ h¯ Vp2/2 λ V = e e h¯ V ∆ˆ (61)

yielding

2 2 i Vˆ iT ( p ? ? p ) i λ ( (q)? ? (q)) e− h¯ H = e− h¯ 2 − 2 e− h¯ V − V (q) iλ ih¯ ih¯ Tp∂q h¯ [ (q 2 ∂p) (q+ 2 ∂p)] (q) fn+1(z)=e− e− V − −V fn (z) . (62)

(q) By using fn (z)=z the transformation for the phase space coordinates is found to be

p = p + λ 0(q ) q = q Tp (63) n+1 n V n+1 n+1 n − n

which is identical to the classical trajectory given by Eq. (59) after redefining pn viap ñ = Tpn. However we will keep the λ and T dependences as they appear in Eq. (63) for reasons to be clarified later. The analogy with the classical trajectories is limited to Eq’s (59) and (63) or to those obtained from Eq’s (59) and (63) by linear canonical transformations. It was discussed in section 2.2 that such transformations create an equivalence class maintaining the covariance under time evolution. It is expected that the time evolution is non-covariant for a general function of phase space. By direct inspection it is observed that polynomials P (z)ofdegreer 2 evolve identically r ≤ 12 in the classical and quantum cases. As a specific example we calculate the one time step quantum evolution of f = p3 q which is of order r =4inz. We find (¯h2) difference between the non-covariant full quantum solution and the covariant classical partO indicated by (q)and (c`) respectively. The difference is

3 (q) 3 (c`) 3 (q) 3 3 2 3 [p q]n+1 [p q]n+1 =[p q]n+1 pn+1 qn+1 = λ h¯ ∂q V (q) (64) − − −2 q=qn+1

3 (c`) 3 where use have been made of [p q]n+1 =(pn+1) qn+1 where the latter are ,i.e. pn+1,qn+1 the same for the classical and the quantum cases. It turns out that a generic difference between the (q)and(c`) transformations of any function is always (¯h2) or higher. One immediate observation in Eq. (64) is that it is no longer possible, evenO after suitable normalizations, to 3 (q) express [p q]n+1 by the parameter κ. The reason is the broken classical covariance under the time evolution. The second observation in Eq. (64) is that, depending on the shape of the potential V (q), the long time behavior of the classical and quantum solutions can be very different. If the third derivative of the potential is oscillatory, like in the standard map, the difference between the long time averages of the classical and quantum solutions in Eq. (64) vanishes. For polynomial potentials the right hand side in Eq. (64) is unbounded. In this case, and for sufficiently large n, the classical and the quantum solutions can differ significantly. Another feature of non covariance is about the time evolution of dynamical systems that are related to each other by nonlinear canonical transformations. Such transformations may help in understanding fundamentally different quantum behavior of the classically equivalent systems. As a typical case under the evolution by the Hamiltonian in (57) consider the gauge transformation4

q Q = q + ap3 ,pP = p (65) 7→ 7→ where a is a real parameter. Because of the nonlinear momentum dependence the quantum evolution of the new pair described by Pn,Qn diﬀers from its classical partner. For (65) only the transformation of Q is nontrivial. It can be calculated as

Q(q) Q(c`) = a [p3](q) [p3](c`) n+1 − n+1 n+1 − n+1 3 (q) 3 2 aλ 3 = a [p ]n+1 pn+1 = h¯ ∂q V (q) . (66) − − 4 q=qn+1

What (66) says is that not only the classical and the quantum solutions, but also the transformed and untransformed quantum solutions diﬀer under Eq. (65).

E. Equivalence to the ?-exponential formalism

The quantum Lie generator formalism as outlined in section 2.2 can become an indispensable tool in the quantum phase space. The equivalence of the Lie generators to the quantization by the ?-exponential can be easily demonstrated. ˆ iγ Vˆ ˆ ˆ Consider the unitary transformation U = e A . Expanding U in power series of and applying the Weyl map ˆr ?...?A we ﬁnd that A A A ⇐⇒ A r A | {z } 13 2 3 iγ (iγ) (iγ) ˆ u = e A =1+iγ + ? + ? ? + ... . (67) U⇐⇒ ? A 2! A A 3! A A A Eq. (67) is the well known ?-exponential8.The?-exponential can be used in the transformation of f as

( 1) iγ iγ f 0(z)=u − ?f?u= e− A ?f(z) ?e A ,?= ? ? (68) ? ? z ≡ q,p This equation involves the ?-product of the ?-exponential and it’s direct calculation is noto- riously difficult in yielding analytically closed forms. It is already difficult to calculate the ?-exponential and it is proven so at the most basic level, the harmonic oscillator8.Using the MLT approach the harmonic oscillator solution can be derived quite effortlessly. The iγ calculation can be found in Ref. [4]. By using (67) and the results in section C.2 e? A? in (68) can be expressed as

iγ iγ ( ?) iγ (ˆzR) e?− A ? = e− A = e− A iγ iγ (? ) iγ (ˆzL) ?e? A = e A = e A (69)

iγ Vˆ If we further use Eq. (25) we observe that Eq. (69) is equivalent to f 0 = e A f as given by (30).

F. Conclusion

In this paper, we have introduced a new phase space approach to Weyl quantization referred to as the Moyal-Lie theory. MLT is basically a corrected Poisson-Lie theory by the manifestation of associativity by the ? product. The new approach is equipped with all other features of classical Hamiltonian vector fields. An exceptional case is the need for a new concept of covariance. It is shown here that the covariance is equipped in the non commutative phase space with a star product. This suggest that one may still be able to define trajectories in the non commutative (quantum) sense. The new theory is formally equivalent to the ?-quantization of Flato et al. and Bayen et al.8 and it may be more appealing from the physical point of view. This is partially due to its aspect which has to do with the algebraic similarities with the classical Hamiltonian vector fields and the phase space Lie transformations. The second aspect is the connection with the theory of quantum canonical transformations in the phase space which is reported elsewhere4.

ACKNOWLEDGMENTS

TH is grateful to C. Zachos (HEP Division, Argonne National Laboratory) for discussions. He also acknowledges the support and the hospitality of the DSAT group as well as the Physics Department of the University of Maryland.

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